Interpolatory methods for H∞\mathcal{H}_\infty model reduction of multi-input/multi-output systems

We develop here a computationally effective approach for producing high-quality $\mathcal{H}_\infty$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg, Beattie, and Gugercin for the single-input/single-output (SISO) setting, which combined ideas originating in interpolatory $\mathcal{H}_2$-optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many large-scale settings. We are able to avoid computationally demanding $\mathcal{H}_\infty$ norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better $\mathcal{H}_\infty$ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation

DELMU: A Deep Learning Approach to Maximising the Utility of Virtualised Millimetre-Wave Backhauls

Advances in network programmability enable operators to 'slice' the physical infrastructure into independent logical networks. By this approach, each network slice aims to accommodate the demands of increasingly diverse services. However, precise allocation of resources to slices across future 5G millimetre-wave backhaul networks, to optimise the total network utility, is challenging. This is because the performance of different services often depends on conflicting requirements, including bandwidth, sensitivity to delay, or the monetary value of the traffic incurred. In this paper, we put forward a general rate utility framework for slicing mm-wave backhaul links, encompassing all known types of service utilities, i.e. logarithmic, sigmoid, polynomial, and linear. We then introduce DELMU, a deep learning solution that tackles the complexity of optimising non-convex objective functions built upon arbitrary combinations of such utilities. Specifically, by employing a stack of convolutional blocks, DELMU can learn correlations between traffic demands and achievable optimal rate assignments. We further regulate the inferences made by the neural network through a simple 'sanity check' routine, which guarantees both flow rate admissibility within the network's capacity region and minimum service levels. The proposed method can be trained within minutes, following which it computes rate allocations that match those obtained with state-of-the-art global optimisation algorithms, yet orders of magnitude faster. This confirms the applicability of DELMU to highly dynamic traffic regimes and we demonstrate up to 62% network utility gains over a baseline greedy approach.Comment: remove LaTeX remains in abstract; change the font for acrony